Optimum Steady State Position, Velocity, and Acceleration Estimation Using Noisy Sampled Correspondence Position Data A one-dimensional tracking filter based on the Kalman filtering techniques for tracking of a dynamic target such as an aircraft is discussed. The target is assumed to be moving with constant acceleration and is acted upon by a plant noise which perturbs its constant acceleration motion. The plant noise accounts for maneuvers and/or other random factors. Analytical results for estimating optimum steady state position, velocity, and acceleration of the target are obtained. INTRODUCTION Consider an aircraft moving with constant acceleration and acted upon by a plant noise perturbing its constant acceleration motion. Each position coordinate of the aircraft is assumed to be measured by a radar at uniform sampling intervals of time T. For each position coordinate of the vehicle, the dynamics may be represented as Xn+ X-n + xn T + inT2/2 + anT3/6 xi? = ,XI? +Xn T + anT2/2 += xn + a7 T (1) where xn, xn, and x,> are the vehicle position, velocity, and acceleration at scan n. The plant noise an is (change in acceleration) at scan n, and T is the sampling interval. It is assumed that the plant noise is a random constant between successive observations having zero mean and also uncorrelated with its values at other intervals as in [1]. Equations (1) may be represented in the vector matrix form as Xn+, = FXXn + Gaan (2) with nX_ 1 T T2/2 -T316- Xn= LX F = K 1 T G = T2/2 The measurement equation may be written as x,n(n) = HXn + Vn (3) where xn is the measured position at scan n, v,n the measurement noise at scan n, and H = [1 0 0]. The statistical properties of the measurement noise are also assumed to be the same as in [1]. By the application of Manuscript received February 5, 1987; revised April 16, 1987. 0018-9251/87/0900-0705 $1.00 C 1987 IEEE IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-23, NO. 5 SEPTEMBER 1987 705
Transcript
Optimum Steady State Position, Velocity, andAcceleration Estimation Using Noisy Sampled
Correspondence Position Data
A one-dimensional tracking filter based on the Kalman filteringtechniques for tracking of a dynamic target such as an aircraft isdiscussed. The target is assumed to be moving with constant
acceleration and is acted upon by a plant noise which perturbs itsconstant acceleration motion. The plant noise accounts formaneuvers and/or other random factors. Analytical results for
estimating optimum steady state position, velocity, and accelerationof the target are obtained.
INTRODUCTION
Consider an aircraft moving with constant accelerationand acted upon by a plant noise perturbing its constantacceleration motion. Each position coordinate of theaircraft is assumed to be measured by a radar at uniformsampling intervals of time T. For each position coordinateof the vehicle, the dynamics may be represented as
Xn+ X-n+ xn T + inT2/2 + anT3/6
xi? = ,XI? +Xn T + anT2/2
+= xn + a7 T (1)
where xn, xn, and x,> are the vehicle position, velocity,and acceleration at scan n. The plant noise an is (changein acceleration) at scan n, and T is the sampling interval.It is assumed that the plant noise is a random constantbetween successive observations having zero mean andalso uncorrelated with its values at other intervals as in[1]. Equations (1) may be represented in the vectormatrix form as
Xn+, = FXXn + Gaan (2)
with
nX_ 1 T T2/2 -T316-Xn= LX F = K1 T G = T2/2
The measurement equation may be written as
x,n(n) = HXn + Vn (3)
where xn is the measured position at scan n, v,n themeasurement noise at scan n, and H = [1 0 0]. Thestatistical properties of the measurement noise are alsoassumed to be the same as in [1]. By the application of
Manuscript received February 5, 1987; revised April 16, 1987.
0018-9251/87/0900-0705 $1.00 C 1987 IEEE
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-23, NO. 5 SEPTEMBER 1987 705
Kalman filtering techniques, the optimal estimates of thestate vector are given by
Xn= Xn +± Kn(xm(n)-HXn) (4)
Xn-1 (5)
Kn P, HT (H PnHT +R) -1 (6)
Pn+l =F Pn FT + GQGT (7)
Pn = (I-Kn H) Pn (8)
where X(X,n) is the optimum estimate of the state vectorafter (before) the measurement xm(n) is processed, Pn(Pn)is the covariance matrix of the estimation errors after(before) processing the measurement, Kn is the Kalmangain matrix, R = (JX is the variance of the measurementnoise and Q = a' is the variance of the plant noise.
STEADY STATE ANALYSIS
In the steady state Pn + 1 = Pn = P, Pn + 1 = Pn =P, and K,+, = Kn = K, and hence from (7) and (8) wehave
Y13 = P131(axO7aT)
Y22 = 12P22/(CraT4)
Y23 = 2 P23/(oaT3)
y33 =P33l(aT) (16)
After considerable algebraic calculations, the solution to(10)-(15) may be found as (see Appendix)
= (S +S)/r
V11 = 2S Y13/r
112 =S2 l3/A
Y22 = S2(3+2S+AS2)/(2A) -
Y33 = (S2 +A)/(2A)
Y23 = y332 (17)
where
S, SQRT(S2 + r2)
P - GQGT = F(l-KH)PFT. (9)
If the covariance matrix P is defined as
P1l1 P12 P13P= P12 P22 P23
LP13 P23 P33-
then (9) results in the following six scalar equations
From (8), the elements of P matrix may be found as
Y13 = (SI -S)/rA A
(14) Y,1 = 2S Y13/rA A
(15) Y12 = S2 Y13/AA
Y22 = S2(3 + 2S - AS2)/(2A) - S,
Y33 = (S2 -A) / (2A)
= 2 - 2 (18)
where YVi are as defineu in (16) by replacing (~) by (A).All quantities within the square root (SQRT) signs in (17)and (18) are positive for r - 0. From (6) the elements ofthe Kalman gain matrix may be found as
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-23, NO. 5 SEPTEMBER 1987706
CONCLUSION
K2 = 2AS2(S1 - S)/ (Tr2)
K3 = 12(S1 - S)/(T2 r2).
The mean square values of the ripple in position,velocity, and acceleration are given by [1]
(P 1 - P1P) /oJ = 4S21r2
(P22 - P22)/(oJ2 T4) = (4S - 1)/12
(P33 - P33)/(oFa T2) = 1.
INTERPRETATION OF RESULTS
Analytical results for tracking of an aircraft withconstant acceleration motion and perturbed by a zero-
(19) mean plant noise are obtained analogous to those obtainedfor the constant velocity case presented in [1] in a one-dimensional dynamic model.
Making use of these properties, the two-dimensionaland three-dimensional tracking filters may easily bedeveloped as in [2] and [3].
APPENDIX
The solution to (10)-(15) is obtained as follows.From (14) and (15), we get
From (17)-(19), it is seen that all the filter parametersare expressed in terms of the quantity r = 12 O-x/ (a,, T3).This quantity is a dimensionless parameter and may beregarded as noise-to-signal ratio, a, is the sensor standarddeviation (in feet), and oraT3/6 is the position errorcaused by a constant change in acceleration of c7a ft/s
Graphs of PJ1/CY and P1f/o versus noise-to-signalratio r are shown in Fig. 1. From Fig. 1 it is seen thatP1l/cox is always less than unity whereas P X/ry can be
tI
1-\
-i-
cm
z
0c0j
LIJ
02:
z
,-X
CX
C):CLcl
10 ,00 ,000
ECOISE - TQ- StGNAL RATIO Y - 2 v/o ,|3
Fig. 1. Position accuracy before and after position detennination.
larger or smaller than unity depending upon the randomchange in acceleration or the sampling time or both. It isseen that for r = 287.62, the value of P,/uj.2 is less thanunity. Hence for
T < 0.3468 a,x/QJit is possible to keep the position estimation error belowthe inherent sensor error even just before observations aremade when the error is the largest. It is interesting tonote that at the crossover point of r = 287.62, the valueof Pll/x is 0.5 as in [1].
Y12 + Y13 =2 Yl13 Y33.
From (13), (14), and (15), we get
y- -Y23 = y33 .
From (12), (Al), (A2), and (15), we get, aftersimplification,~~~~~~~~~Yll= 2 S(S1 + S)/r2
= (S1 + S)/rwhere
S = 3 Y2 - 3 Y33 + 1.
From (12) and (14),
12 = S2 Y13/A.From (A1) and (A6)
Y33 = (S2 + A)/(2A).
From (1 1), we have
Y22 = 6 Y33 Y33 -rY13.
(Al)
(A2)
(A3)
(A4)
(A5)
(A6)
(A7)
(A8)
Using the above results we get from (10)
S4 - 6 S3 + 10 S2 - 6 m S + n = 0
where
m = 1 + 2 r2
n = 1 + 3 r2.
(A9)
Solving the biquadratic (A9), the value of S isdetermined; Y22 and Y23 are determined using (A2) and(A8); S,, S2, and A are as defined by (17). Hence thecomplete solution is obtained.
ACKNOWLEDGMENTS
The author expresses his thanks to Dr. R.P. Shenoy,Director, and also to Mr. N.P. Ramasubba Rao, Deputy
Director, Electronics and Radar DevelopmentEstablishments, Bangalore, 560075, India, for theirencouragement and permission to publish this work.
K.V. RAMACHANDRA
Radar C DivisionElectronics and Radar Development EstblishmentDRDO Complex, ByrasandraBangalore 560075India
REFERENCES
[1] Friedland, B. (1973)Optimum steady state position and velocity estimation usingnoisy sampled position data.IEEE Transactions on Aerospace and Electronic Systems,AES-9, 6 (Nov. 1973), 906-911.
[2] Castella, F.R., and Dunnebacke, F.G. (1974)Analytical results for the X, Y Kalman tracking filter.IEEE Transactions on Aerospace and Electronic Systems,AES-10, 6 (Nov. 1974), 891-895.
[3] Ramachandra, K.V., and Srinivasan, V.S. (1977)Steady state results for the X, Y, Z Kalman tracking filter.IEEE Transactions on Aerospace and Electronic Systems,AES-13, 4 (July 1977), 419-423.
A Method of Constructing a Pair ofPseudorandom Sequences With a UsefulCrosscorrelation Function
The synthesizing of pseudorandom signals which possess an oddcrosscorrelation function of useful characteristics is presented. Thecharacteristics are applicable for signal tracking systems such asthose associated with ranging instrumentation and spread-spectrumcommunication. Another property of these signals is that theypossess a zero dc component which may be applied for radiatinguseful carrierless signals.
1. INTRODUCTION
There exist several applications in system engineeringwhere it should be useful to synthesize pairs of wellconstructed signals which optimize the ability of the userto estimate the mutual positioning in time of the signals[1-7]. To meet this property of the signals theircrosscorrelation function should have the followingcharacteristics. 1) It exhibits time sidelobes of minimalsize. 2) The mainlobe consists of two antisymmetricparts. 3) The two antisymmetric extremum points of themainlobe are connected by relatively steep linear region.
The synthesis and analysis of a pair of signals whichpossess the latter properties is presented. Binary signalsare employed for this purpose since the digital technologyis efficiently developed to deal with such signals. Theclass of signals relevant to the present discussion is,however, broader than that of binary sequences, since onemay also assume applications where a nonbinary carrier ismodulated by binary signals synthesized from sequencesdealt with here.
11. METHOD OF SIGNALS CONSTRUCTION
Let x denote one period of an m-sequence, see [7],and let T denote the left shift operator defined in [8]. Thenew sequences constructed can be expressed as w1 -
[x,x] and w2 = [x, -Tix] where i = ± 1, ±2. Thecrosscorrelation function for these two sequences can beshown to be p [x] (j) + p [x, -Tix] (j) = p [x] (j) -
p[x] (j + i) from [8, eq. (2.7)]. Since the sequence ofvalues of unnormalized autocorrelation function Np[x](j) is as follows
j -2 -1 0 1 2
Np[x] (j) -1 -1 N -1 -1Np[x] (j+1) -1 N -1 -1 -1Np[x] (j+2) N -1 -1 -1 -1Np[x](j-1) -1 -1 -1 N -1Np[x] (j-2) -1 -1 -1 -1 N
then the unnormalized crosscorrelation function of wl andw2 has values
casel i-= 0 -1-N 1+N 0 0caseII i=2 -1-N 0 1+N 0 0case I i =i- 1 0 0 1+N -1-N 0case IV i=-2 0 0 1 +N 0 -1-N
depending on the value of i from [8, eq. (1.11)]. Thecrosscorrelation function for the two pulse trains (derivedfrom w1 and w2, respectively) consists of a set of straightlines with endpoints given above, and hence is as shownin Fig. 1. The demonstrated properties of thecrosscorrelation functions shown in Fig. 1 are evidentlythose expected in Section I where an antisymmetricnarrow central lobe was anticipated as well as acancellation of the ripple.
The values of i considered in the previous discussionare limited to i = + 1, + 2. This limitation is due to thefact that for other values of i property 1 (in Section I),cannot be attained.
Ill. DISCUSSION AND CONCLUSION
A method of constructing pairs of binary sequenceswith useful properties has been explained. The mainproperty is due to the attainment of crosscorrelationfunction of the useful form shown in Fig. 1. There exist
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-23, NO. 5 SEPTEMBER 1987708
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